16 must be the product of one more than the exponents in the prime factorization of M.
So M must have one of the following forms.
(i) p^15
(ii) p^7*q
(iii) p^3*q^3
(iv) p^3*q*r
(v) p*q*r*s
Since D(6)=18=2*3^2 we can rule out (i) and (v). Since 18 itself has 6 factors, 2^2 can't be a factor so this rules out (iii).
(ii) would require the number to be 2*3^7 whose first 9 factors are
1,2,3,6,9,18,27,54,81. 8154=27 not the required 17 so (ii) is out.
So the number must be of the form (iv). Specifically 2*3^2*r where r is a prime > 18.
We need this prime to slot into the above list either before or after the 54 and give the difference 17.
It turns out that both options work
5437=17 and 7154=17.
So M is either
2*3^3*37=1998
or
2*3^3*71=3834
Edited on January 27, 2016, 10:21 am

Posted by Jer
on 20160127 10:18:51 